标题： 二维$q$-状态时钟模型在流表示中的对偶性和BKT相变的蒙特卡罗研究 显示英文标题

标题： A Monte Carlo study of the duality and BKT phase transitions of the two-dimensional $q$-state clock model in flow representations

摘要： 二维（2D）$q$状态时钟模型对于$q \geq 5$在温度降低时会发生两次伯仁斯基-科尔蒂茨-托尔斯（BKT）相变。在这里，我们报告了对平方晶格时钟模型在一对流表示中的大范围虫型模拟，分别来自高温和低温展开，对于$q=5$到 9。通过磁化率类似量的有限尺寸标度分析，我们以比现有结果更高的精度确定了临界点。由于双流表示，临界区域中的每个点都被观察到同时表现出一对异常维数，在两次 BKT 相变中分别为（$\eta_1=1/4$，$\eta_2 = 4/q^2$）。 此外，由严格条件定义的近似自对偶点$\beta_{\rm sd}(L)$，即两种流表示中的类似磁化率量相等，被发现几乎与系统大小$L$无关，并在大-$q$极限下表现出$\beta_{\rm sd} \simeq q/2\pi$的行为。 指数$\eta$在$\beta_{\rm sd}$处与$1/q$在统计误差范围内一致，只要$q \geq 5$。我们的工作提供了对二维时钟模型对偶性和自对偶性相关丰富现象的生动演示。 显示英文摘要

摘要： The two-dimensional (2D) $q$-state clock model for $q \geq 5$ undergoes two Berenzinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for $q=5$ to 9 in a pair of flow representations, respectively from the high- and low-temperature expansions. By finite-size scaling analysis of susceptibility-like quantities, we determine the critical points with a precision improving over the existing results. Thanks to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ($\eta_1=1/4$, $\eta_2 = 4/q^2$) at the two BKT transitions. Further, the approximate self-dual points $\beta_{\rm sd}(L)$, defined by the stringent condition that the susceptibility-like quantities in both flow representations are identical, are found to be nearly independent of system size $L$ and behave as $\beta_{\rm sd} \simeq q/2\pi$ asymptotically at the large-$q$ limit. The exponent $\eta$ at $\beta_{\rm sd}$ is consistent with $1/q$ within statistical errors as long as $q \geq 5$. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.